The logarithmic derived category

As a scheme undergoes a simple normal crossing degeneration its coherent sheaves - and their derived category - behave unpredictably. In this talk I will describe a proposal for a category of logarithmic (quasi-)coherent sheaves associated to such a degeneration. The corresponding derived category exhibits Grothendieck-Neeman duality with the expected dualizing complex, and has a perfect diagonal (the categorical signature of smoothness).  Structures on the logarithmic derived category of the central fibre most closely resemble the smooth situation when considered relative a tropical derived category. This category is compatible with Maulik and Ranganathan’s logarithmic Donaldson—Thomas theory and has an associated (logarithmic) moduli stack of logarithmic coherent sheaves.

 

Based on joint work with De Deyn, Dell, Hu, Manali-Rahul, and Schimpf, and on separate joint work with Poiret, and Song for the moduli stack.